In the first part of this master’s thesis, a convexity of functions of one variable is discussed. Following the definition and basic facts of convexity, the concept of Jensen’s inequality is introduced, followed by the continuity and differentiability of convex functions, where also a connection between the convexity and a second derivative of the function is proved. Then a concept of semi-convexity and quasiconvexity is introduced. The first part is concluded by presentation of the Γ function in connection with convexity. In the second part of thesis, the convexity of functions of more variables, or better, the convexity in a finite dimensional vector space is discussed. The emphasis is on the connection between convex sets and convex functions, where the properties of both are connected. The last section deals again with Jensen’s inequality and the other inequalities, derived from it.
|
